**Course co-ordinator(s):** Prof Jennie C Hansen (Edinburgh).

**Aims:**

The aims of this course are

- To develop discrete probability models for data
- To understand important features of these models

**Summary:**

This course provides an introduction to some of the probability models for inference. The main topics covered are:

- Probability models: sample spaces and events; probability measures and their properties; simple probability calculations.
- Conditioning and Independence: conditional probability; `Chain Rule’ for computing probabilities; Bayes’ Theorem; Partition Theorem; independence of events; probability calculations using concepts of conditioning and independence.
- Random variables and their distributions, including the Binomial, Geometric, Poisson, Hypergeometric, and Indicator variables.
- Expectation and standard deviation of a random variable: definitions and properties of expectation and standard deviation.

## Detailed Information

**Pre-requisites:** none.

**Location: **Edinburgh.

**Semester: **2.

**Syllabus:**

1. Introduction to discrete probability models

- Models for random experiments:
- Sample spaces: events, set theory, Venn diagrams, de Morgan’s laws
- Probability functions and the axioms of probability: Choice of probability function, consequences of the axioms.

- Conditional probability and independence of events:
- Definitions of P(A|B) and the definition of independence
- `Chain rule’ for P(A intersection B)
- Bayes Theorem
- Partition Theorem and `tree diagrams’.

2. Special probability models for certain random experiments

- Simple equally likely models
- Sampling without replacement from a finite population:
- ordered samples: model assumptions, calculations and relevant counting arguments
- unordered samples: model assumptions, calculations and relevant counting arguments

- Building models for a sequence of independent sub
- experiments:
- General model assumptions
- Sampling with replacement: calculations and relevant counting arguments
- Bernoulli trials and Binomial models: calculations and relevant counting arguments
- Geometric models
- Other examples

3. Discrete random variables

- Definition of a discrete random variable
- Probability mass functions for discrete random variables
- Expectation and the properties of the expectation of a random variable.
- Variance and the properties of the variance of a random variable.
- Indicator variables

4. Special examples of discrete random variables.

- Binomial and Bernoulli variables
- Geometric and Negative Binomial variables
- Poisson variables
- Hypergeometric variables
- Indicator variables

**Learning Outcomes: Subject Mastery**

At the end of this course, students should be able to

- Carry out probability calculations for basic discrete probability models using standard techniques including the Partition Theorem, Bayes’ Theorem, and the Chain Rule.
- Work out the distribution, expectation, and standard deviation for a discrete random variable that describes the outcome of a random experiment.
- Identify an appropriate standard random variable to describe the outcome of a random experiment.
- Use the properties of expectation and indicator variables to work out the expected value of a counting variable.

**Reading list:**

*A First Course in Probability* by S.M.Ross, 7th ed. (Pearson 2006).

**Assessment Methods:**

2-hour final exam in May (80%) and continuous assessment (2 assignments) (20%)

**SCQF Level: **7.

**Credits:** 15.

## Other Information

**Help:** If you have any problems or questions
regarding the course, you are encouraged to contact the lecturer

**VISION:** further information and course materials
are available on VISION